The two goals of this project were to build a magnetic oscillating train and to model the apparatus with differential equations. Obstacles that had to be overcome in the design and construction of the magnetic train/cart were the inherent attractive-repulsive properties of magnets and minimization of friction. In the modeling process, the group had to understand the principles of physics to accurately represent every force in the equation. Also, the group had to determine which forces and terms were essential to the behavior of the equation.
In the apparatus constructed, the train floats (approximately) ¼" above a magnetic track. Plexiglass walls maintain the train's position over the magnetic tracks. The magnetic train tracks are glued to a 4 foot long 2"x 4" piece of wood. They are placed 1.5" apart and ¾" away the sides. The cart is a ¼" x 2"x 6" piece of wood. The sides of the cart were sanded down slightly to prevent excessive rubbing against the walls. To create the oscillations, the entire apparatus was tilted at different angles. The end of the track was equipped with a neodymium magnet and each end of the cart had two 3/8" thick magnets (for balance) so that when the track was tilted, the cart's end magnet and the track's neodymium magnet would oppose each other. The oscillations continued until the magnetic forces balanced at the equilibrium.
Initially, a design in which air resistance was the only energy sink was desired. Most of the designs tested involved placing magnets underneath the cart and along the Plexiglass walls (and on the sides of the cart). After many trials, the group hypothesized that attempting to magnetically levitate an object in static equilibrium is impossible. The group does not yet know the principles and physics behind this theory.
In addition to the principle hypothesized above, the properties of magnets also prevented the construction of a levitating train. The magnetic field of magnets curves into and out of poles, and the curvature is more pronounced in smaller magnets where the poles are close to each other. To float one magnet above another, the upper magnet must be placed directly above the area on the lower magnet where the field is perpendicular to both surfaces. However, the small size of most magnets (and the proximity of the poles) reduces this region's size, causing such balancing to be extremely difficult. Even if an object could be levitated above this small region, any slight shift in its position would cause an uneven amount of force to be applied, resulting in an acceleration away from equilibrium. Theoretically, such levitation would be possible if a magnet with extremely large cross-sectional area was used. An extremely large magnet's magnetic field would not experience much bending at the center of the surface since it is far from the poles (it would begin to bend at a large distance away from the surface). Consequently, the apparatus had to be equipped with walls holding the cart in place; this balancing act could not be performed with magnets and the crude materials (i.e. wood and tape) available. With the addition of the walls, friction was an inevitable obstacle.
To minimize friction, the sides of the carts were sanded down into a concave shape so that there were only 2 contact points on each side of the cart. Then, Scotch tape was used to cover the wood at the contact points. The Plexiglass was also lined with Scotch tape. Finally, vegetable oil was placed on the Scotch tape to further reduce friction.
In order to provide the strongest repulsion possible, a Neodymium-Iron-Boron magnetic disc (7/8" in diameter) was placed on top of the other two magnets in the repulsion unit. The two rectangular magnets (½"x 1"x 2") hold the neodymium magnet in place, and prevent it from flipping. The two rectangular magnets are embedded in foam. The repulsion unit rests against a block of wood bolted at the end of the track. Once the apparatus had been built, the data was collected using a sonar sensor. The sonar sensor was placed at the open end of the track (opposite end from the repulsion unit), and the track was tilted to different angles/heights. The cart was then pulled up and released with the sonar recording the cart's position versus time.
For this section, please download our paper
Here is our equation:
and the magnetic constant, k=.0016037
the mass of the cart, m = .2930412 kg
We also found the Lyapunov function for our system:
The plot of the Lyapunov function looks like
Analysis
Though the data coincided well with the theoretical model, there are slight discrepancies that can be accounted for by modifying the equation. Even though the terms for air viscosity and air resistance were discarded to simplify the numerical analysis, they are necessary to accurately model the motion. By adding more damping to the equation (from air interaction), the time interval for which the oscillations occur will be decreased. A slight decrease in the length of the time interval is necessary since the time intervals obtained from the theoretical model were all greater than the intervals obtained from the physical apparatus (see Table 1, above). The Original and Modified Equations were discarded because Mathematica was having difficulty numerically computing the solution.
A second modification would be to attempt to standardize the friction between the cart and the walls. Even though a friction constant of 0.355 was employed to maintain a standard equation, the physical apparatus was not inspected in any way to ensure that the friction was approximately the same for every trial. Since the lubricant employed (vegetable oil) is a liquid, the amount of liquid per unit area varies along the length of the Scotch tape on the walls. Though a small amount of vegetable oil enhances the cart's ability to overcome friction, an excess of oil would actually increase drag on the cart. Consequently, a method of application to maintain a constant amount of liquid per unit area is needed. Methods for consideration include, but are not limited to: brushing the oil on, or steadily dripping the oil onto the strip. Also, the group should consider alternative lubricants (i.e. graphite) to help minimize friction. More accurate measuring methods would also assist in creating a more precise model. For example, a Gauss meter to measure the strength of the magnetic field would assist in finding the correct magnetic constant, k.
The presence of friction affected the final equilibrium position measured. As the cart would come to rest, static friction would arise, and there were now three forces holding the cart in its equilibrium position (rather than two): friction, magnetic repulsion, and the weight of the cart. The static friction prevented the exact same equilibrium from being attained for the same tilt angle; the position would vary by approximately 1.5 cm between trials. Thus, the data obtained for the theoretical and actual final positions was extremely accurate (given the presence of friction) since they differed from each other by, at most, 1.534 cm.
Conclusion
In conclusion, the magnetic oscillating train (perfectly levitating in static equilibrium) that the group desired to build is an impossible feat. The equilibrium attained from such levitation is always dynamic; there are slight fluctuations in the position of the cart as the system's energy oscillates. After this fact was acknowledged, the group screwed Plexiglass walls into the sides of the track to prevent the cart from being pushed out. Using differential equations and physics, the group successfully modeled the oscillations created when the entire cart-track apparatus was tilted to various angles. While there were slight discrepancies between the theoretical and actual data, they can be attributed largely to the inconsistencies arising from friction.
Email us with any question, comments, or concerns:
gtg690b@prism.gatech.edu